On bivalent semantics and natural deduction for some infectious logics

Logic Journal of the IGPL 30 (1):186-210 (2022)
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Abstract

In this work, we propose a variant of so-called informational semantics, a technique elaborated by Voishvillo, for two infectious logics, Deutsch’s |${\mathbf{S}_{\mathbf{fde}}}$| and Szmuc’s |$\mathbf{dS}_{\mathbf{fde}}$|⁠. We show how the machinery of informational semantics can be effectively used to analyse truth and falsity conditions of disjunction and conjunction. Using this technique, it is possible to claim that disjunction and conjunction can be rightfully regarded as such, a claim which was disputed in the recent literature. Both |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| are formalized in terms of natural deduction. This allows us to solve several problems: to develop a natural deduction calculus for |${\mathbf{S}_{\mathbf{fde}}}$| containing the standard form of disjunction elimination (in contrast to the calculus by Petrukhin), to introduce the first natural deduction calculus for |$\mathbf{dS}_{\mathbf{fde}}$| and to reflect the fundamental symmetry between |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| on proof-theoretical level forming a convenient basis for obtaining their well-known extensions |$\mathbf{K}^{\mathbf{w}}_{\mathbf{3}}$| and |$\mathbf{PWK}$|⁠.

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Alexander Belikov
Moscow State University

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References found in this work

The logic of paradox.Graham Priest - 1979 - Journal of Philosophical Logic 8 (1):219 - 241.
What is a Theory of Meaning? (II).Michael Dummett - 1976 - In Gareth Evans & John McDowell (eds.), Truth and Meaning: Essays in Semantics. Oxford: Clarendon Press.
Paraconsistent logic.Graham Priest - 2008 - Stanford Encyclopedia of Philosophy.
What is a theory of meaning?Michael A. E. Dummett - 1975 - In Samuel Guttenplan (ed.), Mind and Language. Oxford University Press.

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